Basic Operational Laws Research Articles

Hesitant Bipolar-Valued Neutrosophic Set: Formulation, Theory and Application

This paper proposes a hesitant bipolar-valued neutrosophic set (HBVNS) based on the combination of bipolar neutrosophic sets and hesitant fuzzy sets. The proposed set generalizes the notions of fuzzy set, intuitionistic fuzzy set, hesitant fuzzy set, single-valued neutrosophic set, single-valued neutrosophic hesitant fuzzy set, bipolar fuzzy set and bipolar neutrosophic set. Further, we define the basic operational laws, union, intersection and complement for hesitant bipolar-valued neutrosophic elements (HBVNEs) and study its associated properties. Some relevant examples are also given to provide a better understanding of the proposed concept. Two aggregation operators are developed based on HBVNS which are the hesitant bipolar-valued neutrosophic weighted averaging (HBVNWA) and the hesitant bipolar-valued neutrosophic weighted geometric (HBVNWG). A decision making method is developed based on new sets and the proposed HBVNWA and HBVNWG operators. Finally, an illustrative example is given to show the applicability of the proposed decision making method. A comparative analysis with the existing methods is also provided.

Intuitionistic Fuzzy Double Integrals and Their Fundamental Properties

Intuitionistic fuzzy set (IFS), as a significant generalization of the fuzzy set (FS), takes the membership function and the non-membership function into consideration simultaneously. The core components of an IFS are the ordered pairs which are called intuitionistic fuzzy numbers (IFNs). More recently, the intuitionistic fuzzy calculus (IFC) has been established, which is based on the basic operational laws of IFNs. As far as the IFC with several variables is concerned, its theory is still in the initial stage. The purpose of this paper is to systematically establish the theory of the intuitionistic fuzzy double integrals (IFDIs). To accomplish this, we first construct the IFDIs, and then assign them concrete values. To understand the IFDIs in depth, we investigate their fundamental properties in detail and offer simple proofs for them. Next, we give a counterexample to verify that in the IFC the first and second mean value theorems do not hold, and interpret the reason from the perspective of the geometric figure. In addition, we compare the results in the IFC with those in the classical calculus to demonstrate that they are thoroughly different despite their forms being sometimes similar. Finally, we offer an application to show the utility of the proposed IFDIs.

Multiple attribute group decision-making method with single-valued neutrosophic interval number information

ABSTRACTTo describe decision makers’ judgments for an evaluated object more comprehensively in the decision-making process, this paper firstly defines the concept of a single-valued neutrosophic interval number (SVNIN), which is composed of both its interval number (interval judgment) and its single-valued neutrosophic number (SVNN) (confident degree). Then we present the basic operational laws of SVNINs and the score function of SVNIN based on the attitude index and confident level of decision makers to compare SVNINs. Next, we propose a single-valued neutrosophic interval number weighted arithmetic averaging (SVNINWAA) operator and a single-valued neutrosophic interval number weighted geometric averaging (SVNINWGA) operator to aggregate SVNIN information. Further, a multiple attribute group decision-making (MDGDM) method using the proposed SVNINWAA or SVNINWGA operator is established under an SVNIN environment. Finally, an illustrative example regarding the selection problem of global suppliers is presented to show the application and effectiveness of the developed method.

Fermatean fuzzy linguistic set and its application in multicriteria decision making

In this paper, we propose the concept of Fermatean fuzzy linguistic term sets based on linguistic term sets and Fermatean fuzzy sets. The basic operational laws, the score function, and the accuracy function of Fermatean fuzzy linguistic numbers are provided. Then we propose the Fermatean fuzzy linguistic weighted aggregation operator, the Fermatean fuzzy linguistic weighted geometric operator, and the Fermatean fuzzy linguistic distance measures. Furthermore, we extend the TOPSIS method to the proposed distance measures, and the ranking of alternatives is obtained under a Fermatean fuzzy linguistic environment. An example is provided to illustrate the feasibility and effectiveness of the proposed method, and a comparison with the existing method is also analyzed. Finally, the sensitivity analysis of the parameter in the proposed distance measure is also discussed.

Cosine Distance Measure between Neutrosophic Hesitant Fuzzy Linguistic Sets and Its Application in Multiple Criteria Decision Making

This paper proposes a neutrosophic hesitant fuzzy linguistic term set (NHFLTS) based on hesitant fuzzy linguistic term set (HFLTS) and neutrosophic set (NS), which can express the inconsistent and uncertainty information flexibly in multiple criteria decision making problems. The basic operational laws of NHFLTS based on linguistic scale function are also discussed. Then we propose the generalized neutrosophic hesitant fuzzy linguistic distance measure and discuss its properties. Furthermore, a new similarity measure of NHFLTS combines the generalized neutrosophic hesitant fuzzy linguistic distance measure and the cosine function is given. A corresponding cosine distance measure between NHFLTSs is proposed according to the relationship between the similarity measure and the distance measure, and we develop the technique for order preference by similarity to an ideal solution (TOPSIS) method to the obtained cosine distance measure. The main advantages of the proposed NHFLTS is defined on linguistic scale function, the decision makers can flexibly convert the linguistic information to semantic values, and the proposed cosine distance measure between NHFLTSs with TOPSIS method can deal with the related decision information not only from the point of view of algebra, but also from the point of view of geometry. Finally, the reasonableness and effectiveness of the proposed method is demonstrated by the illustrative example, which is also compared to the other existing methods.

Dealer using a new trapezoidal cubic hesitant fuzzy TOPSIS method and application to group decision-making program

In this paper, we describe the idea of trapezoidal cubic hesitant fuzzy number. We discuss some basic operational laws of trapezoidal cubic hesitant fuzzy number and hamming distance of trapezoidal cubic hesitant fuzzy numbers (TrCHFNs). We introduce the new concept of the trapezoidal cubic hesitant fuzzy TOPSIS method. Furthermore, we extend the classical trapezoidal cubic hesitant fuzzy TOPSIS method to solve the MCDM method based on the trapezoidal cubic hesitant fuzzy TOPSIS method. The new ranking method for TrCHFNs is used to rank the alternatives. Finally, an illustrative example is given to verify and prove the practicality and effectiveness of the proposed method. We compared the proposed method to the existing methods, which shows the trapezoidal cubic hesitant Fuzzy TOPSIS method is more flexible to deal uncertainties and fuzziness. The main advantage of these operators is that it is to provide more accurate and significant results.

Linguistic Pythagorean fuzzy sets and its applications in multiattribute decision-making process

In this article, a new linguistic Pythagorean fuzzy set (LPFS) is presented by combining the concepts of a Pythagorean fuzzy set and linguistic fuzzy set. LPFS is a better way to deal with the uncertain and imprecise information in decision making, which is characterized by linguistic membership and nonmembership degrees. Some of the basic operational laws, score, and accuracy functions are defined to compare the two or more linguistic Pythagorean fuzzy numbers and their properties are investigated in detail. Based on the norm operations, some series of the linguistic Pythagorean weighted averaging and geometric aggregation operators, named as linguistic Pythagorean fuzzy weighted average and geometric, ordered weighted average and geometric with linguistic Pythagorean fuzzy information are proposed. Furthermore, a multiattribute decision-making method is established based on these operators. Finally, an illustrative example is used to illustrate the applicability and validity of the proposed approach and compare the results with the existing methods to show the effectiveness of it.

Derivatives and differentials for multiplicative intuitionistic fuzzy information

By using the unsymmetrical scale instead of the symmetrical scale, the multiplicative intuitionistic fuzzy sets (MIFSs) reflect our intuition more objectively. Each element in a MIFS is expressed by an ordered pair which is called a multiplicative intuitionistic fuzzy number (MIFN) and is based on the unbalanced scale (i.e., Saaty’s 1-9 scale). In order to describe the derivatives and differentials for multiplicative intuitionistic fuzzy information more comprehensively, in this paper, we firstly propose two new basic operational laws for MIFNs, which are the subtraction law and the division law. Secondly, we describe the change values of MIFNs when considering them as variables, classify these change values based on the basic operational laws for MIFNs, and depict the convergences of sequences of MIFNs by the subtraction and division laws. Finally, we focus on the multiplicative intuitionistic fuzzy functions and derive some basic results related to their continuities, derivatives and differentials, and also give their application in selecting the configuration of a computer.

Multiple Attribute Group Decision-Making Method Based on Linguistic Neutrosophic Numbers

Existing intuitionistic linguistic variables can describe the linguistic information of both the truth/membership and falsity/non-membership degrees, but it cannot represent the indeterminate and inconsistent linguistic information. To deal with the issue, this paper originally proposes the concept of a linguistic neutrosophic number (LNN), which is characterized independently by the truth, indeterminacy, and falsity linguistic variables. Then, we define the basic operational laws of LNNs and the score and accuracy functions of LNN for comparing LNNs. Next, we develop an LNN-weighted arithmetic averaging (LNNWAA) operator and an LNN-weighted geometric averaging (LNNWGA) operator to aggregate LNN information and investigate their properties. Further, a multiple attribute group decision-making method based on the proposed LNNWAA or LNNWGA operator is established under LNN environment. Finally, an illustrative example about selecting problems of investment alternatives is presented to demonstrate the application and effectiveness of the developed approach.

Aggregation operators of neutrosophic linguistic numbers for multiple attribute group decision making

Based on the concept of neutrosophic linguistic numbers (NLNs) in symbolic neutrosophic theory presented by Smarandache in 2015, the paper firstly proposes basic operational laws of NLNs and the expected value of a NLN to rank NLNs. Then, we propose the NLN weighted arithmetic average (NLNWAA) and NLN weighted geometric average (NLNWGA) operators and discuss their properties.

Subtraction and Division Operations of Simplified Neutrosophic Sets

A simplified neutrosophic set is characterized by a truth-membership function, an indeterminacy-membership function, and a falsity-membership function, which is a subclass of the neutrosophic set and contains the concepts of an interval neutrosophic set and a single valued neutrosophic set. It is a powerful structure in expressing indeterminate and inconsistent information. However, there has only been one paper until now—to the best of my knowledge—on the subtraction and division operators in the basic operational laws of neutrosophic single-valued numbers defined in existing literature. Therefore, this paper proposes subtraction operation and division operation for simplified neutrosophic sets, including single valued neutrosophic sets and interval neutrosophic sets respectively, under some constrained conditions to form the integral theoretical framework of simplified neutrosophic sets. In addition, we give numerical examples to illustrate the defined operations. The subtraction and division operations are very important in many practical applications, such as decision making and image processing.

Aggregation operators of neutrosophic linguistic numbers for multiple attribute group decision making.

Based on the concept of neutrosophic linguistic numbers (NLNs) in symbolic neutrosophic theory presented by Smarandache in 2015, the paper firstly proposes basic operational laws of NLNs and the expected value of a NLN to rank NLNs. Then, we propose the NLN weighted arithmetic average (NLNWAA) and NLN weighted geometric average (NLNWGA) operators and discuss their properties. Further, we establish a multiple attribute group decision-making (MAGDM) method by using the NLNWAA and NLNWGA operators under NLN environment. Finally, an illustrative example on a decision-making problem of manufacturing alternatives in the flexible manufacturing system is given to show the application of the proposed MAGDM method.

Novel basic operational laws for linguistic terms, hesitant fuzzy linguistic term sets and probabilistic linguistic term sets

In the process of decision making, people sometimes may feel more comfortable to express their preferences by linguistic terms instead of the quantitative form. However, as the basic premise of operations, the existing operational laws of linguistic terms and the extended linguistic term sets are very unreasonable. In order to overcome this issue, in this paper, we redefine some more logical operational laws for linguistic terms, hesitant fuzzy linguistic elements (HFLEs) and probabilistic linguistic term sets (PLTSs) based on two equivalent transformation functions. These novel operational laws can not only avoid the operation values exceeding the bounds of LTSs, but also keep the operation results more reasonable in decision making with linguistic information. Furthermore, the operational laws can keep the probability information complete when computing with PLTSs. Additionally, lots of properties of the operational laws are discussed, and some three-dimensional figures are drawn to show the regions of different operational laws of linguistic terms more vividly.

Probabilistic linguistic term sets in multi-attribute group decision making

When expressing preferences in qualitative setting, several possible linguistic terms with different weights (represented by probabilities) may be considered at the same time. The probabilistic distribution is usually hard to be provided completely and ignorance may exist. In this paper, we first propose a novel concept called probabilistic linguistic term set (PLTS) to serve as an extension of the existing tools. Then we put forward some basic operational laws and aggregation operators for PLTSs. After that, we develop an extended TOPSIS method and an aggregation-based method respectively for multi-attribute group decision making (MAGDM) with probabilistic linguistic information, and apply them to a practical case concerning strategy initiatives. Finally, the strengths and weaknesses of our methods are clarified by comparing them with some similar techniques.

Model for evaluating the virtual enterprise’s risk with 2-tuple linguistic information

With the increasingly fierce competition in the market, the virtual enterprises are becoming a realistic choice of enterprises. As a new business mode, the virtual enterprises bring tremendous opportunities to improve the competitiveness of enterprises to a certain degree. However, without enough practical and proper theoretical experience, the failure probability of virtual enterprises is high. The virtual enterprises have certain advantages, but the complexity of their organization makes the uncertainties increasing, the risk analysis control is difficult and the risk management is complex. In order to achieve smooth operation and expected profits, the virtual enterprises must succeed to avoid the risks. Therefore, the risk management for virtual enterprises is significant in theoretical and practical research. In this paper, we investigate the 2-tuple linguistic multiple attribute group decision making problems with incomplete weight information, some basic concepts and operational laws of 2-tuple linguistic variables are introduced. A model based on the entropy weight method, by which the attribute weights can be determined, is established. According to the traditional ideas of grey relational analysis (GRA), the optimal alternative(s) is determined by calculating the linguistic degree of grey relation of every alternative and 2-tuple linguistic positive ideal solution and 2-tuple linguistic negative ideal solution. It is based on the concept that the optimal alternative should have the largest degree of grey relation from positive ideal solution and the smallest degree of grey relation from the negative ideal solution. Finally, an illustrative example for evaluating the virtual enterprise's risk is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Exponential operations for intuitionistic fuzzy numbers and interval numbers in multi-attribute decision making

Intuitionistic fuzzy numbers (IFNs) have already been applied to many fields, especially in multi-attribute decision making (MADM). Based on the basic operational laws and information aggregation methods of IFNs, MADM with intuitionistic fuzzy information has become more and more popular. In this paper, we investigate the MADM problems where the attribute values take the form of interval numbers and the weight information on the attributes are expressed as IFNs. We first propose a novel exponential operational law based on IFNs and interval numbers, and then study some of its desirable properties. Based on the exponential operational law, we put forward an intuitionistic fuzzy weighted exponential aggregation operator, and utilize it to develop a MADM method. Finally, we apply our method to solve the decision making problem under uncertainty.

Multiple attribute group decision making based on generalized power aggregation operators under interval-valued dual hesitant fuzzy linguistic environment

This paper aims to investigate the type of fuzzy multiple attribute group decision making (MAGDM) where arguments being aggregated are allowed to support each other. In order to enable decision makers to express their preferences more comprehensively, we firstly put forward a hybrid tool, an interval-valued dual hesitant fuzzy linguistic set (IVDHFLS), which employs interval-valued hesitant membership and nonmembership degrees to assess linguistic terms. Basic operational laws for IVDHFLS are discussed, also a distance measure is designed to overcome irrationality in traditional methodology for hesitant fuzzy sets, i.e., artificially adding values to mismatching membership or nonmembership degrees. We next develop fundamental generalized power average aggregation operators for IVDHFLS, including power average operator, power geometric average operator, power ordered weighted average operator and power ordered weighted geometric average operator. Desirable properties and special cases of these aggregation operators are further analyzed. Furthermore, based on the generalized operators above, we construct two approaches for MAGDM with mutually supportive arguments being aggregated under interval-valued dual hesitant fuzzy linguistic environments. Finally, case studies are conducted to verify effectiveness and practicality of the developed approaches.

A consensus reaching model for 2-tuple linguistic multiple attribute group decision making with incomplete weight information

The aim of this paper is to put forward a consensus reaching method for multi-attribute group decision-making (MAGDM) problems with linguistic information, in which the weight information of experts and attributes is unknown. First, some basic concepts and operational laws of 2-tuple linguistic label are introduced. Then, a grey relational analysis method and a maximising deviation method are proposed to calculate the incomplete weight information of experts and attributes respectively. To eliminate the conflict in the group, a weight-updating model is employed to derive the weights of experts based on their contribution to the consensus reaching process. After conflict elimination, the final group preference can be obtained which will give the ranking of the alternatives. The model can effectively avoid information distortion which is occurred regularly in the linguistic information processing. Finally, an illustrative example is given to illustrate the application of the proposed method and comparative analysis with the existing methods are offered to show the advantages of the proposed method.

Multi-attribute group decision making using combined ranking value under interval type-2 fuzzy environment

In this paper, we investigate a new method to handle multiple attribute group decision making (MAGDM) problems based on combined ranking value under interval type-2 fuzzy environment, in which all the attribute values provided by experts take the form of interval type-2 fuzzy sets (IT2FSs). We first introduce some basic concepts and related operational laws on IT2FSs. Then, we put forward three kinds of ranking value formulas to calculate the ranking value of IT2FSs based on arithmetic average (AA) operator, geometric average (GA) operator and harmonic average (HA) operator, respectively, and discuss some of its desirable properties. Based on these properties, we define the concept of combined ranking value and also further develop a new interval type-2 fuzzy entropy with trigonometric sine function to measure the uncertainty of the IT2FSs. By using the three types ranking value formulas and interval type-2 fuzzy entropy we proposed, a new approach based on the principle of combinatorial optimization with ranking-entropy and the least squares for determining attribute weight is given. Furthermore, a decision making procedure based on combined ranking value is given to select the best alternative(s). Finally, a simple practical example concerns that urban rail transit evaluation is provided to illustrate the practicality and effectiveness of the proposed method, and a comparative analysis is performed.

Grey relational analysis method for 2-tuple linguistic multiple attribute group decision making with incomplete weight information

AbstractWith respect to 2-tuple linguistic multiple attribute group decision making problems with incomplete weight information, some basic concepts and operational laws of 2-tuple linguistic variables are introduced. An optimization model based on the maximizing deviation method, by which the attribute weights can be determined, is established. According to the traditional ideas of grey relational analysis (GRA), the optimal alternative(s) is determined by calculating the linguistic degree of grey relation of every alternative and 2-tuple linguistic positive ideal solution and 2-tuple linguistic negative ideal solution. It is based on the concept that the optimal alternative should have the largest degree of grey relation from positive ideal solution and the smallest degree of grey relation from the negative ideal solution. The method has exact characteristic in linguistic information processing. It avoided information distortion and losing which occur formerly in the linguistic information processing. Finally, a numerical example is used to illustrate the use of the proposed method. The result shows the approach is simple, effective and easy to calculate.